complex analysis - Definition of a simply connected region

I am reading Bak and Newman's Complex Analysis, and I am having a difficult time understanding the definition of a simply connected region. Here's the definition:

A region $D$ is simply connected if its complement is “connected within $\epsilon$ to $\infty$.” That is, if for any $z_0 \in D^{c}$ and $\epsilon > 0$, there is a continuous curve $\gamma (t), 0 \leq t < \infty$, such that:

i) $d(\gamma(t), D^c) < \epsilon$ for all $t \geq 0$,

ii) $\gamma(0) = z_0$,

iii) $\lim_{t \rightarrow \infty } \gamma(t) = \infty$.

While I understand that, intuitively, the last two conditions state that $D^c$ is unbounded in the sense that any point in the complement can be "joined to $\infty$" using a line/curve that lies within $D^c$. What about the first condition? Also, it'd be great if someone could motivate this definition as well, without invoking algebraic toploogical notions.